120 anticipated approach, we calculated the variance and covariance matrix in
Table 6-10
as follows.
a ˆ
bˆ
ˆ
a ˆ
3.43E-10 -3.23E-10
-9.78E-09 bˆ
-3.23E-10 3.11E-10
-1.86E-07
ˆ -9.78E-09
-1.86E-07 2.52E-04
Table 6-10: Variance and covariance results
The variance diagonal of each parameter shows a small value, which implies that our estimation is not far from its mean, while the covariance is small, implying that the
parameters are independent of each other.
6.9 Check the Goodness-Of-Fit to Data
The remaining datasets were then used to check the fit of our model to the data. We aimed to predict the residual life of the diesel engines given the observed information.
Two datasets that illustrate typical monitoring information are presented in
Figure 6-11
below.
Figure 6-11: Typical monitoring observation
121 Engine 83000128 showed a steady increase in monitored information and failed at
27,500 hours, while engine 83000130 had similar readings at the early stages but showed a sharp increase beginning at 14,300 hours of operation and failed at 22,200
hours. The same
i
y
on the earlier stages is because they are extrapolated from other engines as the earlier information for both engines is not available to us. This evidence
supports our assumption that the higher the total metal concentration the shorter the residual time.
Two examples of the prediction model of residual time in terms of probability distributions and actual residual times are shown in
Figure 6-12
and
Figure 6-13
below.
Figure 6-12: Case 1 – pdf and actual residual time with monitoring information for engine
83000128
122
Figure 6-13: Case 2 – pdf and actual residual time with monitoring information for engine
83000130
Both results show that the actual residual time is in the predicted region. To test the hypothesis that the monitored information does influence the model prediction, we
calculated the pdf. of the residual time of the diesel engine without the observed information as a comparison. The formulation for the residual time that depends on age
without condition-monitoring data can be written using
x p
, as at time
i
t
1 1
i i
i i
i i
i
dx t
t x
p t
t x
p x
p
1 1
i t
x i
i t
x i
i
dx e
t x
e t
x
i i
i i
6-36
The actual residual time was also plotted. Results are shown in
Figure 6-14
below.
123
Figure 6-14: pdf and actual residual time without monitoring information for engine 83000128
To make a more straightforward judgement, we plotted the pdf of each distribution at the last observation point before a failure occurred. A comparison of both approaches is
illustrated in Figure 6-15 and Figure 6-16 below.
Figure 6-15: Case 1 – pdf of residual time of 83000128 at last observation point
124
Figure 6-16: Case 2 – pdf of residual time of 83000130 at last observation point
Both cases show a significant difference but at different scales, which reflects the
influence of monitoring data. Figure 6-15 shows that the increment of
i
y
for engine 83000128 is small, which indicates that the influence of condition-monitoring
information is small. In contrast, Figure 6-16 shows a very significant difference
between the pdf. of residual life with and without monitoring information. This is because the higher increment of
i
y
does have some effect on the residual time. This shows that the monitoring information plays an important role in prediction.
6.10 Testing the Model
